Mostly Surfaces (Student Mathematical Library) by Richard Evan Schwartz

by Richard Evan Schwartz

This book presents a number of topics related to surfaces, such as Euclidean, spherical and hyperbolic geometry, the fundamental group, universal covering surfaces, Riemannian manifolds, the Gauss-Bonnet Theorem, and the Riemann mapping theorem. The main idea is to get to some interesting mathematics without too much formality. The book also includes some material only tangentially related to surfaces, such as the Cauchy Rigidity Theorem, the Dehn Dissection Theorem, and the Banach-Tarski Theorem. The goal of the book is to present a tapestry of ideas from various areas of mathematics in a clear and rigorous yet informal and friendly way. Prerequisites include undergraduate courses in real analysis and in linear algebra, and some knowledge of complex analysis.

Paperback Bunko

First published June 1, 2011

About the author

Richard Schwartz grew up in Los Angeles. He wore only blue clothes between the ages of 7 and 11. He spent his youth obsessively playing tennis until video games distracted him. He majored in math at UCLA, got a PhD in math from Princeton, and is currently the Chancellor's Professor of Mathematics at Brown University. His research interests lie in geometry and dynamics. He likes to do mathematical experiments on the computer and then find proofs for the results he discovers.

Rich was an Invited Speaker at the 2002 International Congress of Mathematicians, a Guggenheim Fellow in 2003, a Clay Research Scholar in 2009, and Simons Fellow in 2012. He is the author of a number of books, including Spherical CR Geometry and Dehn Surgery, Outer Billiards on Kites, You Can Count on Monsters, Man Versus Dog, Unnecessary Surgery, and The Extra Toaster. He lives with his wife and two daughters in Barrington, Rhode Island. In his spare time, he listens to music, writes comic books, cycles on the bike path near his house, walks on the beach, or spends time with his family.

Reviews

[Justin · Rating 1 out of 5]

Perhaps my greatest criticism of the book is that it was written as if the reader already understands the topics being discussed. For example, the section on isomorphism theorem was vague. Although the explanation may be understandable to someone who understands the topic, obviously if one is consulting the book, it is most likely due to a desire to understand something that is currently not understood.

The book was mathematically nonrigorous (i.e. it did not show every step). So often when an explanation was needed, the book would leave it as an exercise to the reader—very frustrating. The ends of chapters, however, do not include problems to help check for understanding.

A small redeeming quality, though, is that it gave good intuitions to topology and had helpful illustrations.

Perhaps Schwartz is a great mathematician but a great teacher through writing he is not.